called a capacitary measure
on the (compact) set \(K\) . A capacitary measure \(\gamma _K\)  is in general non-unique, and it
has the following properties [1]}:If Frostman's maximum principle holds (Sect. REF ), the capacitary measures \(\gamma _K\)  are also called equilibrium measures on \(K\) , for their potentials take the constant value 1 for nearly all \(x\in K\) . (In fact, it follows from () that \(\kappa \gamma _K\leqslant 1\)  on \(X\) , which together with () gives \(\kappa \gamma _K=1\)  n.e. on \(K\) .)
\(&\gamma _K(X)=\kappa (\gamma _K,\gamma _K)=c(K),\\&\kappa \gamma _K\geqslant 1\text{ \ n.e.\ on\ }K,\\&\kappa \gamma _K\leqslant 1\text{ \ on\ }S(\gamma _K),\\&\kappa \gamma _K=1\text{ \ $\gamma _K$-a.e.\ on $X$.}\) 
